ssfa {ssfa}

Spatial stochastic frontier estimation

Description

This function estimates the Spatial Stochastic Frontier model introduced by Fusco and Vidoli (2013) in the following form:

log(y_{i}) = log(f(x_{i};\beta_i)) +v_{i}-u_{i}

u_{i}=\rho \sum_{i}w_{i.}u_{i} + \widetilde{u_{i}}

where y_i are the outputs, x_i the inputs, v_i the stochastic noise, u_{i} the inefficiency term, rho the spatial lag, w_{i.} a standardized row of the spatial weights matrix and \widetilde{u_{i}} the stochastic noise of the inefficiency term.

Usage

ssfa(formula, data = NULL, data_w = NULL, intercept = TRUE, pars = NULL, par_rho = TRUE, 
                 form = "cost")

Arguments

formula

an object of class formula (or one that can be coerced to that class): a symbolic description of the model to be fitted.

data

an optional data frame containing the variables in the model.

data_w

a data frame containing the spatial weight matrix.

intercept

logical. If true the model includes intercept.

pars

initial values for the parameters to be estimated.

par_rho

logical. If true the function estimates the Spatial Stochastic Frontier (SSFA) otherwise the classical Stochastic Frontier (SFA).

form

specifies the form of the frontier model as "cost" or "production".

Value

ssfa returns the following objects of class ssfa:

y

the dependent variable.

x

the covariates.

X

the model matrix.

coef

the estimated coefficients.

sc

the form of the frontier model estimated (-1 = cost, 1 = production).

hess

a symmetric matrix giving an estimate of the Hessian at the solution found.

logLik

the value of the log likelihood function.

ols

the linear model for the LR-test.

sigmau2

the estimation of sigmau2 (only if par_rho=FALSE): value of inefficiency variance.

sigmau2_dmu

the estimation of sigmau2_dmu (only if par_rho=TRUE): value of the part of the inefficiency variance due to DMU's specificities.

sigmau2_sar

the estimation of sigmau2_sar: value of the part of the inefficiency variance due to the spatial correlation.

sigmav2

the estimation of sigmav2: value of the stochastic error variance.

sigma2

the estimation of sigma2: value of the total variance.

rho

the estimation of the spatial lag parameter rho.

fun

the distribution of the inefficiency term u.

list_w

a listw object from nb2listw (See nb2listw).

Note

NOTE 1: In this version the distribution of the inefficiency term u is only "half-normal".

NOTE 2: The method used to maximize the log likelihood function is the Newton-Raphson. Please see the R function maxNR of the maxLik package for details (Henningsen and Toomet (2011)).

NOTE 3: Please note that the classical SFA inefficiency variance sigmau2, in the SSFA, is decomposed into sigmau2_dmu and sigmau2_sar, respectively the part of inefficiency variance due to DMU's specificities and to the spatial dependence, i.e. sigmau2 = sigmau2_dmu + sigmau2_sar and consequently the total variance is given by sigma2 = sigmau2_dmu + sigmau2_sar + sigmav2.

Author(s)

Fusco E. and Vidoli F.

References

Battese, G. E., and T. J. Coelli (1995). A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data. Empirical Economics 20(2): 325-332.

Fusco, E. and Vidoli, F. (2013). Spatial stochastic frontier models: controlling spatial global and local heterogeneity, International Review of Applied Economics, 27(5) 679-694.

Fusco, E. (2020). Spatial Dependence in Efficiency Parametric Models: A Generalization and Simulation Studies, "Scienze Regionali, Italian Journal of Regional Science" Speciale/2021, 595-618.

Kumbhakar, S. C., and C. A. K. Lovell (2000). Stochastic Frontier Analysis, Cambridge University Press.

Henningsen, A. and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458.

Examples

library(ssfa) 
data(SSFA_example_data)
data(Italian_W)
ssfa <- ssfa(log_y ~ log_x, data = SSFA_example_data, 
             data_w=Italian_W, form = "production", par_rho=TRUE)

### SSFA total variance decomposition 
sigma2 = ssfa$sigmau2_dmu + ssfa$sigmau2_sar + ssfa$sigmav2
sigma2
ssfa$sigma2